During the manufacture of an electric motor armature, an armature shaft and lamination assembly machine is used to press an armature shaft into an aperture in the middle of a stack of slotted laminations made from iron or other metal and that form the core of the armature. Armature shafts are normally formed as solid round rods so that there is typically no concern with regard to the rotational position of the armature shaft about its axis. Certain armature shafts have integrally-formed eccentric cam bodies extending axially from one of their ends. To assist in balancing such an armature shaft to avoid excessive vibration during use, it can be useful to control the rotational orientation of the armature shaft relative to the rotational orientation of the stack of laminations. A prior method and apparatus used to rotationally orient armature shafts having an eccentric cam body extending from one end thereof is found in commonly-owned U.S. Pat. No. 5,500,993.